Skip Navigation

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn027, 19 pages, doi:10.1093/imrn/rnn027 published on April 12, 2008
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Lin, Y.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

The Log-Concavity Conjecture for the Duistermaat–Heckman Measure Revisited

Yi Lin

Department of Mathematics, University of Toronto 40, St. George St., Toronto, Ontario M5S 2E4, Canada

Correspondence: Correspondence to be sent to: ylin{at}math.toronto.edu

Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat–Heckman measure: a Hamiltonian six-manifold whose fixed-points set is the disjoint union of two copies of T4. In this article, for any closed symplectic four-manifold N with b+ > 1, we show that there is a Hamiltonian six-manifold M such that its fixed-points set is the disjoint union of two copies of N and such that its Duistermaat–Heckman function is not log-concave. On the other hand, we prove that if there is a torus action of complexity 2 such that all the symplectic reduced spaces taken at regular values satisfy the condition b+ = 1, then its Duistermaat–Heckman function has to be log-concave. As a consequence, we prove the log-concavity conjecture for Hamiltonian circle actions on six manifolds such that the fixed-points sets have no 4-dimensional components, or only have 4-dimensional pieces with b+ = 1.



References

  1. Adem A., Leida J., Ruan Y. Orbifolds and Stringy Topology (2007) Cambridge: Cambridge University Press. Cambridge Tracts in Mathematics, 171.
  2. Atiyah M. Convexity and commuting Hamiltonians. The Bulletin of the London Mathematical Society (1982) 14:1–15.[CrossRef]
  3. Baldridge S. Seriberg–Witten vanishing theorem for S1 manifold with fixed points. Pacific Journal of Mathematics (2004) 217:1–10.[ISI]
  4. Bott R., Tu L. Differential Forms in Algebraic Topology (1982) New York: Springer.
  5. Chen W., Ruan Y. Orbifold Gromov–Witten theory. Orbifolds in Mathematics and Physics (2002) Providence, RI: American Mathematical Society. 25–86. Contemporary Mathematics, 310.
  6. Chen W., Ruan Y. A new cohomology theory of orbifold. Communications in Mathematical Physics (2004) 248(1):1–31.[ISI]
  7. Chern S. S., Hirzebruch F., Serre J.-P. On the index of a fibred manifold. Proceedings of the American Mathematical Society (1957) 8:587–596.[CrossRef]
  8. Duistermaat J. J., Heckman G. J. On the variation in the cohomology of the symplectic form of the reduced phase space. Inventiones Mathematicae (1982) 69:259–268.[CrossRef][ISI]
  9. Fulton W. Introduction to Toric Varieties. (1993) Princeton, NJ: Princeton University Press. Annals of Mathematics Studies, 131.
  10. Godinho L. Blowing up symplectic orbifolds. Annals of Global Analysis and Geometry (2001) 20:117–162.[CrossRef][ISI]
  11. Gompf R. A new construction of symplectic manifolds. Annals of Mathematics (1995) 142:527–598.[CrossRef][ISI]
  12. Graham W. Logarithmic convexity of push-forward measures. Inventiones Mathematicae (1996) 123:315–322.[ISI]
  13. Guillemin V., E. Lerman S., Sternberg. On the Konstant multiplicity formula. Journal of Geometry and Physics (1988) 5:721–750.[CrossRef]
  14. Guillemin V., Sternberg S. Convexity properties of the moment mapping. Inventiones Mathematicae (1982) 67:491–513.[CrossRef][ISI]
  15. Guillemin V., Sternberg S. Symplectic Techniques in Physics. (1984) Cambridge: Cambridge University Press.
  16. Guillemin V., Sternberg S. Birational equivalence in the symplectic category. Inventiones Mathematicae (1989) 97:485–522.[CrossRef][ISI]
  17. Karshon Y. Periodic Hamiltonian flows on four dimensional manifolds. Contact and Symplectic Geometry—Thomas C. B., ed. (1996) Cambridge: Cambridge University Press. 43–47. INI Publications, 8.
  18. Karshon Y. Example of a non-log-concave Duistermaat-Heckman measure. Mathematical Research Letters (1996) 3(4):537–540.
  19. Lerman E., Tolman S. Hamiltonian torus actions on symplectic orbifolds and toric varieties. Transactions of the American Mathematical Society (1997) 349(10):4201–4230.[CrossRef][ISI]
  20. Lin Y. Examples of non-Kähler Hamiltonian circle manifolds with the strong Lefschetz property. Advances in Mathematics (2007) 208(2):699–709.[CrossRef][ISI]
  21. Mcduff D. The moment map for circle actions on symplectic manifolds. Journal of Geometry and Physics (1988) 5(2):149–160.[CrossRef]
  22. Mcduff D., Salamon D. Introduction to Symplectic Topology (1998) Oxford: Oxford University Press. Oxford Mathematical Monographs.
  23. Metzler D. S. A wall crossing formula for the signature of symplectic quotients. Transactions of the American Mathematical Society (2000) 352(8):3495–3521.[CrossRef][ISI]
  24. Ieke M. Orbifolds as groupoids: An introduction. Orbifolds in Mathematics and Physics (2002) Providence, RI: American Mathematical Society. 205–222. Contemporary Mathematics, 310. (Madison, WI, 2001).
  25. Okounkov A. Brunn–Minkowski inequality for multiplicities. Inventiones Mathematicae (1996) 125:405–441.[CrossRef][ISI]
  26. Okounkov A. Log-concavity of multiplicities with an application to characters of U({infty}). Advances in Mathematics (1997) 127(2):258–282.[CrossRef][ISI]
  27. Okounkov A. Multiplicities and Newton polytopes. Krillov's Seminar on Representation Theory (1998) 181. Providence, RI: American Mathematical Society. 231–244. American Mathematical Society Translations, Series 2.
  28. Okounkov A. Why should multiplicities be log-concave. The Orbit Method in Geometry and Physics (2003) Boston, MA: Birkhauser Boston. 329–347. Progress in Mathematics, 213. (Marseille, 2000).
  29. Park B. D., Szabó Z. The geography problem for irreducible spin four-manifolds. Transactions of the American Mathematical Society (2000) 352:3639–3650.[CrossRef][ISI]
  30. Ruan Y. Stringy geometry and topology of orbifolds. Contemporary Mathematics (2002) 312:187–224.
  31. Satake I. On a generalization of a notion of manifold. Proceedings of the National Academy of Sciences of the United States of America (1956) 42:359–363.[Free Full Text]
  32. Satake I. The Gauss–Bonnet theorem for V-manifolds. Journal of the Mathematical Society of Japan (1957) 9:466–492.
  33. Sternberg R. On minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field. Proceedings of the National Academy of Sciences of the United States of America (1977) 74:5235–5254.[Abstract/Free Full Text]
  34. Weinstein A. A universal phase space for the particles in Yang–Mills fields. Letters in Mathematical Physics (1978) 2:417–420.[CrossRef][ISI]
  35. Weinstein A. Symplectic V-manifolds, periodic orbits of Hamiltonian systems and the volume of certain Riemannian manifolds. Communications on Pure and Applied Mathematics (1997) 30:265–271.[CrossRef]
  36. Yan D. Hodge structure on symplectic manifolds. Advances in Mathematics (1996) 120(1):143–154.[CrossRef][ISI]

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Lin, Y.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?